Consider a polygonization of the plane by convex polygons of a given minimal size that meet edge-to-edge and vertex-to-vertex.

What's the “official” name of such a polygonization?

Such polygonizations of the plane induce infinite graphs.

How can such abstract graphs be characterized?

Somehow like this: “A graph is induced by a polygonization of the plane iff it is infinite, planar, 3-vertex-connected, and P.” (The question asks for property P, since infinite, planar and 3-vertex-connected those graphs obviously are.)

Basically, you can construct a circle packing with any prescribed (three-connected) combinatorics. What you lose when you go from finite to infinite is uniqueness, in a spectacular way: it should be true that one can get the carrier of the packing to be any Jordan domain.

@Igor, thank you very much, I'll try to get your paper. For my better understanding: Is it correct, that the infinite planar 3-vertex-connected graphs are induced by polygonizations of the plane as specified above (edge-to-edge, vertex-to-vertex) and the finite planar 3-vertex-connected graphs are induced by polygonizations of the sphere (a.k.a. polyhedra)?
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Hans StrickerNov 30 '12 at 8:55

@Hans: that's one way of thinking of it. As for the paper, you can get it on arxiv in some version...
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Igor RivinDec 1 '12 at 4:43